Issue exporting animation with blend keys to FBX
The exporting option to support your blendshapes (infalible) is to export a .dae
On the panel options check "EXPORT SHAPE KEYS".
Voilá.

My sincere gratitude if you vote me as the answer.

ps: Yes, I also use Autodesk's FBx review. Normally (2016 FBX format) will BRING in all keyshapes in the model (works in Unity as well).

But a blast from the past with .DAE and go long!

BFCT_Schiller